Unit Digit Concept – MKS GMAT App

Units Digit – Core Concept

The units digit is the last digit before the decimal point.

Examples:

Units digit of 79 is 9
Units digit of 76546 is 6

Useful in:

Multiplication of large numbers
Powers and exponentiation
Modulo 10 and 5 operations

Units Digit in Multiplication

To simplify: take only the units digit of each number.

Example:

What is the units digit of
$85945 \times 89 \times 58307$ ?

Use only last digits:
$5 \times 9 \times 7 = 45 \times 7 = 315$ → units digit = 5

Cyclicity of Units Digits

The units digit of powers repeats in cycles. Identify the cycle, divide the exponent by the cycle length, and use the remainder to find the result.

Cyclicity Table

Cycle Length	Digits	Example Powers
1	0, 1, 5, 6	Always same (e.g., $5^n = 5$ )
2	4, 9	$4^1 = 4$ , $4^2 = 6$ , repeat
4	2, 3, 7, 8	Repeats every 4 (see below)

Example: $7^{345}$

Cycle of 7:

$\begin{align*} 7^1 &= 7 \ 7^2 &= 49 \Rightarrow 9 \ 7^3 &= 343 \Rightarrow 3 \ 7^4 &= 2401 \Rightarrow 1 \end{align*}$

Cycle: 7, 9, 3, 1
Now,
$345 \mod 4 = 1$ ⇒ Use first value in cycle → 7

Summary of Exponent Rules

Type	Rule
1-cycle	Units digit = base
2-cycle	Odd exponent → base, even exponent → base²
4-cycle	Divide exponent by 4, use remainder to pick from cycle

Factorials – Trailing Zeros & Prime Powers

Factorials are common in GMAT questions involving:

Trailing zeros
Power of a prime
Divisibility by a composite number

Trailing Zeros in $n!$

Each trailing zero = factor of 10 = $2 \times 5$
Since 2s are plenty, just count 5s:

Formula:

$\left\lfloor \frac{n}{5} \right\rfloor + \left\lfloor \frac{n}{25} \right\rfloor + \left\lfloor \frac{n}{125} \right\rfloor + \cdots$

Example: Zeros in $32!$

$\left\lfloor \frac{32}{5} \right\rfloor + \left\lfloor \frac{32}{25} \right\rfloor = 6 + 1 = \boxed{7}$

Power of a Prime in $n!$

Formula:

$\sum_{k=1}^{\infty} \left\lfloor \frac{n}{p^k} \right\rfloor$

Stop when $p^k > n$

Example: Power of 2 in $25!$

$\left\lfloor \frac{25}{2} \right\rfloor + \left\lfloor \frac{25}{4} \right\rfloor + \left\lfloor \frac{25}{8} \right\rfloor + \left\lfloor \frac{25}{16} \right\rfloor = 12 + 6 + 3 + 1 = \boxed{22}$

Power of Non-Prime in $n!$

Steps:

Prime factorize the number
Find power of each prime in $n!$
Divide each result by exponent in factorization
Take the minimum

Example: Power of 900 in $50!$

Factor: $900 = 2^2 \times 3^2 \times 5^2$

Step 1: Count powers in $50!$

Prime	Calculation	Result
2	$25 + 12 + 6 + 3 + 1$	47
3	$16 + 5 + 1$	22
5	$10 + 2$	12

Step 2: Divide by exponent in 900

Prime	Divide by	Final Power
2	2	$23$
3	2	$11$
5	2	$6$

Final Answer:
$900^6$ divides $50!$

Manual vs Formula Check: $17!$

Prime	Formula	Total
2	$8 + 4 + 2 + 1 = \boxed{15}$	15
3	$5 + 1 = \boxed{6}$	6
5	$3 = \boxed{3}$	3

GMAT Tricks Summary

What You’re Finding	Focus On
Trailing Zeros	Count 5s only
Number of 15s in $n!$	Count 5s (3s are more frequent)
Number of 10s in $n!$	Count 5s
Number of 8s = $2^3$ in $n!$	Count 2s, then divide by 3

Final Takeaway

Use these tools to simplify and speed up:

Units digit: Use cyclicity rules and remainder logic
Factorials: Use prime count formulas and shortcuts
Apply patterns: Recognize them quickly and eliminate wrong choices fast